How to intuitively visualize sets ?

You’re totally fine thinking of a set as a bracketed collection of elements.
For me, it depends on the set, or what I'm doing to it.

For example, if my set is the set of all rational-valued sequences that tend to zero, I imagine a load of them "graphed" in front of me, overlapping. There are no specific examples in there, just a whole mass together.

Then if I want to check whether e.g. any of them can have terms that actually equal zero, I just take one, set a few terms to zero and see what happens. In this case, it still fits nicely within my set.

Or, for example, if my set is some arbitrary class X and I'm constructing its power set, I imagine a bunch of points in 2D space (maybe 15 but I'm not counting, it's just an arbitrary picture) and imagine someone has drawn a loop around e.g. 4 of them that will be in one subset. And then there are lots and lots of these loops, and those loops are the power set.

Don't try to imagine sets as the notation like {1, 3, 5}, as that doesn't get you anywhere unless you have a small, finite set like {1, 3, 5}. Of course, once you get your intuitive ideas by such imagery, you can formalize your ideas using proper set notation, logic etc. rather than these amorphous visual diagrams.
You don’t need to know anything beyond naive set theory to learn real analysis. In naive set theory, we can just assume we can construct a set {X : P(X)} and not run into issues, that is, such an item can exist and is not contradictory. This is of course not true. For example, the power set of a set has strictly greater cardinality. However, if the set of all sets could occur, then such a set would contradict the claim about the cardinality of the power set. Another example is Russel’s Paradox. ZFC is axiomatic because we need a set of rules for a consistent theory. However, you will probably never care about these rules if you want to learn analysis, so you can simply just adopt a naive point of view.
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We just need a way to know if something is in the set or not.  The things in the set comprise the set.
The point is you don’t have enough structure with just sets to do mathematics. Honestly just pick up all analysis textbook